Games and Economic Behavior 43 (2003) 214–238
www.elsevier.com/locate/geb
The effectiveness of English auctions
Zvika Neeman a,b,c
a Department of Economics, Boston University, 270 Bay State Road, Boston, MA 02215, USA 1
b Department of Economics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, Israel 91905
c Center for the Study of Rationality, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel 91904 2
Received 10 April 2001
Abstract
We study the performance of the English auction under different assumptions about the seller’s
degree of “Bayesian sophistication.” We define the effectiveness of an auction as the ratio between
the expected revenue it generates for the seller and the expected valuation of the object to the bidder
with the highest valuation (total surplus). We identify tight lower bounds on the effectiveness of the
English auction for general private-values environments, and for private-values environments where
bidders’ valuations are non-negatively correlated. For example, when the seller faces 12 bidders
who the seller believes have non-negatively correlated valuations whose expectations are at least as
high as 60% of the maximal possible valuation, an English auction with no reserve price generates
an expected price that is more than 80% of the value of the object to the bidder with the highest
valuation.
 2003 Elsevier Science (USA). All rights reserved.
JEL classification: D44; D82
Keywords: Effectiveness; Worst-case; English auction; Second-price auction; Private values model
1. Introduction
The notion of optimality is central to economics. Yet, economic theory typically only
distinguishes between optimal and sub-optimal outcomes. By and large, there is no attempt
to quantify how far from optimality are sub-optimal outcomes, allocations, or institutions.
E-mail address: [email protected].
URL address: http://people.bu.edu/zvika.
1 Between 9.1.2002 and 12.15.2002 and after 8.31.2003.
2 Until 8.31.2002 and between 12.15.2002 and 8.31.2003.
0899-8256/03/$ – see front matter  2003 Elsevier Science (USA). All rights reserved.
doi:10.1016/S0899-8256(03)00005-8
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
215
In this paper we attempt to make a preliminary step toward quantification of optimality
in the context of auction theory. Specifically, we study the performance of the (singleobject) English auction in private-values environments.3 We define the effectiveness of an
auction as the ratio between the expected revenue it generates for the seller and the expected
valuation of the object to the bidder with the highest valuation (which coincides with total
surplus when bidders are risk-neutral or risk-averse).4 We identify tight lower bounds
on the effectiveness of the English auction for several “classes” of environments and
under three different assumptions about the seller’s degree of “Bayesian sophistication”
as expressed in his ability to set an appropriate reserve price. Specifically, we consider
the English auction with no reserve price, with a fixed positive reserve price and with
an optimally chosen reserve price. Our results show that the English auction performs
reasonably well in a wide class of environments. As will become clearer below, they may
be interpreted as quantifying the “optimality” of the English auction, or alternatively, as
establishing the “cost” of relying on the “simple” English auction relative to the “optimal”
auction in those circumstances where the latter extracts the full surplus.
At a perhaps more practical level, in recent years, several governments around the world
have auctioned off parts of the electromagnetic (airwave) spectrum for commercial use
with the main stated objective of promoting efficiency or “putting licenses into the hands
of those who value them most”5 (Milgrom, 1996, Chapter 1, p. 3). While the revenues
obtained exceeded expectations by a factor of ten or more, to the extent that maximizing
revenue is also an important objective, as it is likely to be in private auctions, existing
theory provides no way of assessing the effectiveness of the auction form used in terms
of what fraction of the total sum of bidders’ willingness to pay was obtained. The method
described in this paper provides a first step towards being able to form such assessments.
Generally, the effectiveness of any auction form depends on the environment that is
considered. In any particular environment, the closer effectiveness is to one, the closer
the auction is to extracting the full surplus. We seek to determine the effectiveness of the
English auction in the environment, within a given class of possible environments, in which
it is the lowest. In this sense, we perform worst-case analysis of the performance of the
English auction. Our results illustrate the robustness of the English auction in the following
3 The English auction has many variants. One such variant that is referred to by Milgrom and Weber (1982)
as the Japanese version of the English auction is idealized as follows. Before the auction begins, the bidders are
given the opportunity to inspect the object and realize their valuations. The bidders choose whether to be active
at the start price that is equal to the reserve price set by the seller. As the auctioneer raises the price, bidders drop
out one by one. No bidder who has dropped out can become active again. The auction ends as soon as no more
than one bidder remains active. The remaining bidder gets the object for the prevailing price. If several bidders
dropped out simultaneously, ending the auction, one of these bidders is chosen randomly and gets the object at
the price at which she quit.
In private values environments, the English auction is outcome-equivalent to a modified Vickrey auction, or
a sealed-bid second-price auction where the seller may set a reserve price (see, e.g., (Milgrom and Weber, 1982)).
All of our results therefore apply to second price auctions as well.
4 In the natural sciences, concepts that quantify aspects of the quality of performance are widely used. For
example, the notion of “energetic efficiency” which is defined as “what you get out of some device divided by
what you put in” (Vogel, 1998, p. 156) is similar to the notion of effectiveness presented here.
5 For details, see McAfee and McMillan (1996) and Milgrom (1996).
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Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
sense: A seller who is uncertain about which environment he is facing within a certain class
of environments is guaranteed an expected revenue (as a proportion of total surplus) that
is not lower and in general higher than worst-case effectiveness. Thus, for those classes of
environments for which the worst-case effectiveness of the English auction can be shown
to be “high,” a seller who is uncertain about the environment, is unable to figure out the
optimal auction, and even if he is able, is suspicious about whether bidders understand the
optimal auction’s rules and is doubtful whether they employ Bayesian–Nash equilibrium
strategies, is well advised to employ an English auction; even in the worst-case, his losses
from not doing otherwise will be small.
We establish the following results. We parametrize all possible private-values environments by the number of bidders, n, and their expected valuations of the object, α, as a
percent of the maximum possible valuation. We obtain a lower bound on the ratio between
the expected revenue generated by the English auction and expected total surplus under
three different assumptions on the seller’s behavior:
(1) the seller does not set a reserve price,
(2) the seller sets a fixed reserve price that maximizes his expected revenue given his
beliefs about the expectations of bidders’ valuations for the object, and
(3) the seller sets an optimal reserve price given his belief about the distribution of bidders’
valuations for the object.
These three assumptions can be thought of as corresponding to three different levels of
“Bayesian sophistication.” From a seller that fails to recognize the fact that setting a
positive reserve price may increase his expected revenue, to a seller who recognizes the
usefulness of a reserve price but is unable to articulate a belief about bidders’ valuations
beyond a specification of their expected valuations, to a seller who can fully articulate his
beliefs and set an optimal reserve price accordingly.6 Some readers have pointed to an
apparent tension between our assumption that the seller may be so rational so as to be
capable of setting an optimal reserve price given his beliefs, and our method of analysis
which focuses on a “low rationality” worst-case analysis. We believe that in light of the
fact that even for the simple environments considered in this paper (correlated private
values environments with risk averse-bidders), the problem of identifying the optimal
auction is still very much an open one,7 the decision to employ an English auction
with an optimally chosen reserve price, especially when the worst-case performance of
this auction is reasonable, is very sensible. Moreover, the fact that for private-values
environments with non-negatively correlated bidders’ valuations the differences between
the worst-case performance of the English auction with and without a reserve price is quite
6 We ignore the issue of whether the seller’s beliefs are “correct.” In a Bayesian world, beliefs are subjective.
The best that any Bayesian rational agent can ever do is to maximize with respect to her beliefs.
7 Even for correlated general values environments with risk neutral bidders, where optimal auctions that
succeed in extracting the entire bidders’ surplus have been identified (Crémer and McLean, 1988; McAfee
and Reny, 1992), the optimality of these auctions depends on the controversial assumption that the seller’s and
the bidders’ beliefs are consistent. An assumption that is not needed here. See Neeman (1999) for additional
discussion of this and related points.
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
217
small, suggests that less sophisticated sellers should find employing English auctions (even
without a reserve price) even more sensible.
For each level of the seller’s degree of Bayesian sophistication and every given n and α,
we determine the effectiveness of the English auction in the worst possible private-values
environment. The identified bounds are tight, and we present examples of environments
that attain them. We then repeat this exercise for environments where bidders’ valuations
are non-negatively correlated. We assume, specifically, that bidders’ valuations of the
object are conditionally independent and identically distributed.
As expected, the worst-case effectiveness of the English auction improves as the number
of bidders, n, and their expected valuations for the object, α, increase. Obviously, when
bidders’ valuations are negatively correlated, the possibility of setting an appropriately
chosen reserve price is very valuable. For example, consider an environment with two
bidders who have negatively correlated valuations such that when one bidder’s valuation is
one, the other bidder’s valuation is zero and vice versa. In such an environment, an English
auction with no reserve price generates an expected revenue of zero, but an English auction
with a reserve price of one, generates an expected revenue that is equal to the total surplus,
one. The English auction is more effective when bidders’ valuations are more plausibly
assumed to be non-negatively correlated. For example, when n = 12 and α = 60%, even
in the worst possible case, an English auction with no reserve price sells the object at an
expected price that is more than 80% of the expected value of the object to the bidder with
the highest valuation.
The research that is most closely related to the work reported here is the series of
papers that culminated in the work of Rustichini et al. (1994) (see also the discussion
in Section 6 below). Rustichini et al. demonstrated that the inefficiency of double-auctions
under symmetric equilibria in i.i.d. environments converges to zero at an asymptotic rate
of the order of magnitude of c/(nm) where n is the number of buyers, m is the number
of sellers, and c is a constant that depends on the particular environment considered.
We perform a similar exercise on the English auction. However, instead of considering
efficiency, we focus on seller’s revenues and consider a much wider class of environments
without restricting the set of equilibria. Furthermore, whereas Rustichini et al. only
identified asymptotic rates of convergence, we identify tight lower bounds for any number
of bidders.8 More recently, Satterthwaite and Williams (1999) have shown that the doubleauction is also worst-case asymptotically optimal. That is, there does not exist any other
exchange mechanism that has a faster asymptotic rate of convergence to efficiency.
The rest of the paper is organized as follows. In the next section we present the
model and state the general problem. We present the results for general private-values
environments in Section 3, and the results for private-values environments where bidders’
valuations are non-negatively correlated in Section 4. By focusing on worst rather than, say,
“average” performance, worst-case analysis tends to emphasize the “weakness” rather than
“strength” of a mechanism. We therefore devote Section 5 to analysis of the worst-case
effectiveness of another commonly used sale mechanism—the posted-price mechanism.
8 Recently, Swinkels (1998, 1999) established the asymptotic efficiency of discriminatory auctions and a class
of uniform price auctions for multiple identical goods in private values environment where bidders’ valuations
are independent but there may be some aggregate uncertainty about demand and supply.
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Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
The results compare unfavorably with those of the English auction. The point of the
comparison is not to argue that from the seller’s perspective the English auction is superior
to the posted-price mechanism, we believe that much is obvious, but rather to “calibrate”
the readers’ expectations about what constitutes “reasonable” worst-case performance. We
conclude in Section 6 with an additional discussion of motivation and related literature. All
proofs and a short explanation about our calculations are relegated to Appendix A.
2. The model
We consider general private-values environments. A seller has a single object to sell.
There are n potential buyers (bidders) for the object. The object is worth nothing for the
seller but has a value vi ∈ [0, 1] for bidder i ∈ {1, . . . , n}. The payoff to bidder i from
buying the object at a price p depends only on her valuation for the object vi and the price
paid. We assume that it is given by ui (vi − p) where bidder i’s payoff function, ui : R → R,
is assumed to be increasing and such that ui (0) = 0. The payoff to the bidder when she does
not buy the object (and does not pay) is normalized to zero. We assume that each bidder
knows her valuation for the object. Except for that, we make no other assumption about
the bidders’ beliefs. In particular, we do not assume that a common prior exists, nor that
the buyers’ and seller’s beliefs are consistent.
If the seller had complete information about the bidders’ valuations for the object and
full bargaining power, he could fetch an expected price of
RFB = E max{v1 , . . . , vn }
for the object by selling it at the price max{v1 , . . . , vn } to the bidder with the highest
valuation. Since the seller’s valuation for the object is zero, if in addition we assume that
the buyers are risk-neutral or risk-averse, then RFB describes the maximal expected total
surplus that can be generated by selling the object.
We employ the following notation: we let B : [0, 1]n → [0, 1] denote the environment
facing the seller, or more precisely, the beliefs that the seller would have had about the
environment he is facing had he been a sophisticated Bayesian. We let G : [0, 1] → [0, 1]
denote the corresponding cumulative distribution of max{v1 , . . . , vn } and H : [0, 1] →
[0, 1] denote the corresponding cumulative distribution of the second highest valuation
from among {v1 , . . . , vn }. By a well-known equality (see, e.g., (Shiryaev, 1989, p. 208))
1
RFB (B) =
1 − G(x) dx.
(1)
0
We are interested in obtaining a lower bound on the ratio between the expected revenue
that the seller can obtain for the object when he employs an English auction and the
maximal total surplus as defined above in (1).9 The expected revenue to the seller depends
9 We do not assume that the seller is risk-neutral, however, a very risk-averse seller would obviously not care
much for our results.
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
219
on whether he sets a reserve price for the object or not. We consider three different types
of seller’s behavior that correspond to the three different degrees of the seller’s Bayesian
sophistication as described above: (1) whether it is because the seller cannot articulate any
beliefs about the buyers’ valuations for the object or because the seller does not realize that
setting a positive reserve price may increase the expected revenue generated by the auction,
the seller does not set a reserve price. (2) The seller recognizes the benefit conferred by
setting a positive reserve price, but because he cannot articulate beliefs about buyers’
valuations that are more specific than the buyers’ expected valuations, he sets a reserve
price given his limited beliefs. A lower bound on the seller’s expected revenue in this case
is given by the seller adopting a maxmin approach, namely, choosing a reserve price that
maximizes the expected revenue for the seller in the environment in which it is the lowest.
Finally, (3) the seller sets an optimal reserve price given his beliefs about the distribution
of the bidders’ valuations.
Denote the expected revenue to the seller from employing an English auction with a
reserve price r in the environment B by RE (B, r). Given a vector of bidders’ valuations
v1 , . . . , vn , let x1 , . . . , xn denote the ordered vector of bidders’ valuations where x1 denotes
the largest valuation from among v1 , . . . , vn , x2 denotes the second largest valuation from
among v1 , . . . , vn , . . . , and xn denotes the smallest valuation from among v1 , . . . , vn . In
English auctions, it is a dominant strategy for the bidders to remain active in the auction
until the price equals their valuations for the object. It therefore follows that
RE (B, r) = r Pr(x1 r) + Pr(x2 r)E[x2 − r|x2 r].
That is, the seller obtains a revenue of r whenever the highest valuation of the bidders,
x1 , is larger or equal to r and an additional revenue of x2 − r when the second highest
valuation is also larger or equal to r. Integration by parts yields,
RE (B, r) = r 1 − G(r − ) +
1
1 − H (x) dx
(2)
r
where G(r − ) = limxr G(x).
Define the effectiveness of the English auction with reserve price r in the environment
B by
RE (B, r)
.
RFB (B)
Note that what appears in the denominator of the definition of effectiveness is the
total surplus generated by the sale rather than the expected revenue the seller can obtain
by employing an optimal auction. For the class of environments where the bidders are
risk neutral, have correlated valuations, and hold beliefs that are consistent with those of
the seller’s, Crémer and McLean (1988) and McAfee and Reny (1992) showed that the
optimal auction succeeds in extracting the entire bidders’ surplus. However, for the type
of environments considered here, where the bidders may be risk-averse, and consistency is
not assumed, a characterization of the optimal auction is unavailable. To the extent that in
these more general cases the optimal auction falls short of extracting the entire buyers’
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Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
surplus, our measure of effectiveness under-estimates the effectiveness of the English
auction relative to that of the “optimal” auction.10
As explained above, for a fixed class of environments B, we are interested in identifying
a lower bound on the effectiveness of the English auction under three different assumptions
about the seller’s behavior. Specifically, we ask what is
RE (B, 0)
EB0 ≡ min
,
(3)
B∈B RFB (B)
or the effectiveness of the English auction with no reserve price in the environment B ∈ B
in which it is the lowest;
RE (B, r)
r(n,α)
,
(4)
≡ max min
EB
r∈[0,1] B∈B RFB (B)
or the effectiveness of the English auction with a positive reserve price that cannot be
tailored to suit the specific environment the seller faces, in the environment B ∈ B in which
it is the lowest; and
RE (B, r(B))
r(B)
EB ≡ min
max
,
(5)
RFB (B)
B∈B r(B)∈[0,1]
or the effectiveness of the English auction with a reserve price r(B) that is chosen
optimally given the seller’s beliefs about the buyers’ valuations B, in the environment
B ∈ B in which it is the lowest. Thus, for example, a seller that employs an English
auction with no reserve price is guaranteed an expected revenue that is at least as high
as minB∈B {RE (B, 0)/RFB (B)} of the expected total surplus when facing any environment
in the class B.
Fig. 1. G, H , RFB , and RE (r).
10 The English auction with an optimally chosen reserve price is optimal in the class of i.i.d. private-values
environments with risk-neutral bidders (Myerson, 1981), hence its effectiveness relative to the optimal auction in
such environments is one.
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
221
Our method of proof makes extensive use of geometric arguments and intuition. The
effectiveness of an English auction with reserve price r for seller’s beliefs that induce
distributions G and H of the first and second highest valuations from among v1 , . . . , vn ,
is represented in Fig. 1 by the ratio between the areas abcde (which by (2) is equal
1
to RE (B, r) = r(1 − G(r − )) + r (1 − H (x)) dx) and 0bde (which by (1) is equal to
1
RFB (B) = 0 (1 − G(x)) dx).
The problem of determining the worst-case environment and worst-case effectiveness is
equivalent to the problem of identifying seller’s beliefs that induce a minimal ratio of the
areas abcde and 0bde.
3. General private values environments
Let Bn,α , α ∈ [0, 1], denote the set of cumulative joint distribution functions of n
random variables that obtain their values on the unit interval and have expectations
larger or equal to α. We interpret Bn,α as representing the general class of private-values
environments with n bidders whose expected valuations are larger than or equal to α% of
the maximal possible valuation. The parameter α describes how high, on average, bidders’
valuations are. It may be interpreted as describing the “attractiveness” of the underlying
auction environment as perceived by the seller.
Denote the worst-case effectiveness of the English auction with no reserve price, with
a fixed reserve price that cannot be tailored to the specific environment, and with a reserve
price that is chosen optimally given the seller’s beliefs in the class of environments Bn,α
r(n,α)
r(B)
by En0 (α), En
(α), and En (α), respectively.
Theorem 1. For every n 2 and α ∈ (0, 1],
nα − 1
En0 (α) = max
,0 .
n−1
(6)
Thinking of the bidders’ valuations as n random variables v1 , . . . , vn with expectation
larger or equal to α, the idea of the proof is to identify the joint distribution that induces
the highest possible expectation of the first order statistic from among v1 , . . . , vn , but the
smallest possible expected second order statistic. The proof shows that the environment
that attains the worst-case bound is one where one bidder, which the seller believes is
equally likely to be any one of the bidders, has valuation 1 with probability min{nα, 1} and
valuation 0 otherwise, and all other bidders have valuations max{(nα − 1)/(n − 1), 0}.
Note that according to these beliefs, bidders’ valuations are negatively correlated.
For the case where the seller cannot tailor the reserve price to the specific environment,
we have,
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Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
Theorem 2. For every n 2 and α ∈ (0, 1],
√

r(n, α)n(α − r(n, α))
1 + 4n − 3


,
0
α
,

(1 − r(n, α))(nα − (n − 1)r(n, α))
2n
r(n,α)
(α) =
En
√

1 + 4n − 3

 nα − 1 ,
α 1,
n−1
2n
where
√
√
α(n − (n − 1)α)
1 + 4n − 3
r(n, α) =
when 0 α ,
n−1+α
2n
and
√
nα − 1
1 + 4n − 3
α 1.
r(n, α) ∈ 0,
when
n−1
2n
(7)
(8)
(9)
The idea of the proof is to identify for every possible reserve price r ∈ [0, 1], the joint
distribution that minimizes the effectiveness of the
√English auction for this particular r,
and then to maximize over r. When 0 α (1 + 4n − 3)/(2n), for every reserve price
r < α,11 the distribution that attains worst-case effectiveness is one where with the highest
possible probability given the constraint that bidders’ expected valuations must be larger
or equal to α, the bidder with the highest valuation has a valuation that is either equal
to 1 or just below the reserve price r, and all other bidders have valuations just below
the reserve price r. Maximization of this lowest possible effectiveness over r reveals
that the distribution that attains the worst case bound is one where one bidder, which
the seller believes is equally likely to be any one of the bidders, has valuation 1 with
probability n(α − r(n, α))/(1 − r(n, α)) and valuation just below
√ r(n, α) otherwise, and
all other bidders have valuations just below r(n, α). When (1 + 4n − 3)/(2n) α 1,
it is impossible to ensure that all the bidders’ valuations except the highest one are below
r (nα − 1)/(n − 1) < α, and so worst-case effectiveness is identical to that obtained
when the seller is constrained to set the reserve price equal to zero. In this case, the
ability to set a positive reserve price does not help the seller.
√ The ability to set a fixed
positive reserve price thus helps the seller only when α (1 + 4n − 3)/(2n), or when the
bidders’ expected valuations are small relative to their number. Intuitively, a fixed positive
reserve price helps the seller when n and α are low.
For the case where the seller chooses the reserve price optimally given his beliefs, we
have,
Theorem 3. For every n 2 and α ∈ (0, 1],
nα − β
Enr(Bn ) (α) =
(n − 1)β
where β is the unique solution in the interval [0, 1] to the equation:
nα − β
nα − β
1 − log
= β.
n−1
n−1
11 It is straightforward to verify that for r α worst-case effectiveness is equal to zero.
(10)
(11)
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
223
The idea of the proof is that in the distribution that attains the lowest bound, it must be
that the seller is indifferent between setting an optimal reserve price for the bidder with
the highest valuation while ignoring all other bidders, and not setting any reserve price at
all. Otherwise, as the proof shows, it is possible to change the distribution and decrease
effectiveness. The distribution that attains the worst case bound is one where one bidder,
which the seller believes is equally likely to be any one of the bidders, has a valuation that
is distributed according to a truncated Pareto distribution,
0,
0 x < ε,
Fε (x) = 1 − ε/x, ε x < 1,
(12)
1,
x = 1,
where ε ∈ [0, 1] is such that ε(1 − n1 log(ε)) = α, and all the other bidders have valuations
equal to ε. Observe that, first, the seller believes that every bidder’s expected valuation of
the object is equal to α. Second, more importantly, again, the distribution that attains the
worst-case bound is one where the bidders’ valuations are negatively correlated. Finally,
third, under the distribution that attains the worst-case bound, every reserve price set by the
seller generates the same expected revenue ε for the seller. Obviously, any reserve price r ε yields a revenue equal to the second highest valuation ε. As for reserve prices r ∈ (ε, 1],
with probability Fε (r), the seller does not sell the object (and obtains a revenue of 0), and
with probability 1 − Fε (r), the seller succeeds in selling the object for the price r. The seller’s expected revenue is therefore given by r(1 − Fε (r)), which can be immediately seen to
equal ε for every r ∈ [ε, 1]. Moreover, Fε is the only function that satisfies the property that,
(13)
r 1 − Fε (r) = ε for every r ∈ [ε, 1].
We depict the values of En0 (α), Enr(n,α) (α), and Enr(B) (α), for n = 4, 12, and the limits as
n tends to infinity, in Table 1.12
Table 1
Worst-case effectiveness for general environments
α
E40 (α)
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1
0
0
0
.0667
.2
.33333
.46667
.6
.73333
.86667
1
r(4,α)
E4
(α)
0
.047649
.10286
.16791
.24621
.34315
.46667
.6
.73333
.86667
1
r(B)
E4
(α)
0
.2605
.33136
.39566
.46007
.52768
.60078
.6816
.77274
.87743
1
0 (α)
E12
0
.01818
.12727
.23636
.34545
.45455
.56364
.67273
.78182
.89091
1
r(12,α)
E12
(α)
0
.06531
.14288
.2369
.34545
.45455
.56364
.67273
.78182
.89091
1
r(B)
E12 (α)
0
.28644
.36427
.43339
.501
.5701
.64262
.72018
.80437
.89695
1
r(∞,α)
0 (α), E
E∞
∞
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1
(α)
r(B)
E∞ (α)
0
.30279
.38322
.45373
.52184
.59062
.66189
.7371
.81757
.90468
1
r(n,α)
limits can be shown to be equal to limn→∞ En0 (α) = limn→∞ En
(α) = α, and
r(Bn )
limn→∞ En
(α) = 1/(1 − log(α)), respectively. The difference between these limits and the values of E 0 ,
0 = E r(25,α) , where it is the largest, it is smaller
E r(α) , and E r(Bn ) when n = 25 is already quite small. For E25
25
12 The
than .04.
224
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
Note that worst-case effectiveness stays bounded away from 1 as the number of bidders
tends to infinity even when the seller sets the reserve price optimally given his beliefs.
This apparently counter-intuitive result is due to the fact that on the distributions that
attain the worst-case bounds, the bidders’ valuations are negatively correlated. Each of
the three distributions that attain the worst-case bounds above describes the case of a seller
who believes that he faces n bidders out of which only one, which he cannot identify,
is “serious” and is willing to pay a high price for the object while all the others are
no more than “warm bodies” with relatively low valuations. Even when the seller sets
the reserve price optimally, and can identify the serious bidder, the fact that the serious
bidder’s valuation is distributed according to the truncated Pareto distribution that satisfies
the special property (13) implies that the seller cannot get a high expected revenue from
this bidder. It should not come as a surprise that when there is only one serious bidder,
increasing the number of bidders has a negligible effect on the effectiveness of the English
auction.
4. Private-values environments with non-negatively correlated bidders’ valuations
While the general class of environments, or more precisely, the general class of seller’s
beliefs about the environment, is the appropriate class in some applications, in many other
cases the appropriate class of environments is smaller. In particular, given that assuming
that bidders’ valuations are non-negatively correlated accords well with the qualitative
features of real life auctions (for this reason, it is also the maintained assumption in much
of auction literature), we restrict our attention in this section to this case.
ciid denote the set of cumulative distribution functions that describe
Specifically, let Bn,α
the joint distribution of n random variables that obtain their values on the unit interval,
have expectations larger or equal to α, and are conditionally independent and identically
ciid describes the beliefs of sellers who believe that there
distributed (c.i.i.d.). The set Bn,α
is some unobservable factor that affects all bidders’ valuations in the same way. For
example, bidders’ valuations may depend on the (unobservable to the seller) state of the
economy, or on the “intrinsic worth of the object,” or on both. Any such (possibly multidimensional) unobservable factor introduces positive correlation into the distribution of
bidders’ valuations but, conditional on it, the bidders’ valuations are independently and
identically distributed.13,14
13 By de Finetti’s theorem (see, e.g., (Durrett, 1991, p. 232)) an infinite sequence of random variables is
conditionally i.i.d. if and only if it is exchangeable. Diaconis and Freedman (1980) describe a sense in which
a finite sequence of exchangeable random variables is approximately conditionally i.i.d.
14 Another widely used assumption that implies non-negative correlation in auction models is affiliation
(Milgrom and Weber, 1982). While many examples of distributions of bidders’ valuations are both conditionally
i.i.d and affiliated, the two notions are independent. Examples of conditionally i.i.d random variables that are not
affiliated are easy to construct; for an example of affiliated random variables that are not conditionally i.i.d., see
(Shaked, 1979, p. 72 (ii)).
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
225
ciid may be represented as a mixture of i.i.d. distributions
Thus, every environment B ∈ Bn,α
in the following way,
n
B(v1 , . . . , vn ) =
F (vi | z) dZ(z)
(14)
i=1
where for every z ∈ Z, F (· | z) ∈ B1,α(z) for some α(z) ∈ (0, 1] and α(z) dZ(z) α.
Denote the worst-case effectiveness of the English auction with no reserve price, with
a reserve price that cannot be tailored to the specific environment, and with a reserve
price that is chosen optimally given the seller’s beliefs in the class of conditionally i.i.d.
ciid by E 0,ciid (α), E r(n,α),ciid (α), and E r(B),ciid (α), respectively. We have
environments Bn,α
n
n
n
the following result.
0,ciid
Theorem 4. For every
(α) is obtained on joint distributions of the form
n α ∈ (0, 1], En
ciid where for every z, F (· | z) ∈ B
B(v1 , . . . , vn ) =
F
(v
|
z)
dZ(z)
∈ Bn,α
i
1,α(z) is a
i=1
“two-step” distribution function of the form:

0, x ∈ (−∞, 0),


p, x ∈ [0, b),
(15)
F (x) =

 q, x ∈ [b, 1),
1, x ∈ [1, ∞),
for some 0 p q 1 and 0 b 1, and Enr(n,α),ciid (α) is obtained on a limit of
a sequence of such distributions.
The idea of the proof is to show that unless every F (· | z) is a two-step function with
support on at most three points 0, b, and 1, the environment B can be changed so as
to decrease effectiveness. In fact, we conjecture, a conjecture that is confirmed by our
numerical analysis but which we cannot prove, that worst-case effectiveness is obtained
on a mixture of two-step distribution functions where the “first step” is always equal
to 0 (i.e., p = 0 in (15) above from which it follows that the distribution is supported
by only two points, b and 1). Intuitively, when the reserve price is constrained to be zero,
conditional on z, under such distributions the expectation of the highest of the bidders’
valuation is “maximized,” whereas the expectation of the second highest of the bidders’
valuations is “minimized.” Remarkably, for the case where the reserve price is constrained
to be zero, our numerical analysis reveals that worst-case effectiveness is obtained on
degenerate mixtures of two-step distribution functions, namely, on i.i.d. distributions of
two-step functions. Again, we conjecture that this is generally the case, but we cannot prove
it. Even remarkably still, our numerical analysis also reveals that the ability to set a fixed
positive reserve price does not allow the seller to obtain higher worst-case effectiveness,
En0,ciid (α) seems to be equal to Enr(n,α),ciid (α) for every value of n 2 and α ∈ (0, 1].
We depict the computed values of En0,ciid (α) = Enr(n,α),ciid (α) for n = 4, 12, and 25 in
Table 2 (see Appendix A for a short explanation on how the calculation was performed).
As suggested by Table 2, even when the seller sets no reserve price, worst-case
effectiveness converges to 1 for every α ∈ (0, 1].
226
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
Table 2
Worst-case effectiveness for conditionally i.i.d. environments
α
r(4,α),ciid
E40,ciid (α), E4
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1
r(12,α),ciid
r(25,α),ciid
0,ciid
E12
(α), E12
0,ciid
E25
(α), E25
0
.32411
.47259
.57934
.66507
.73769
.80117
.85785
.90925
.95638
1
0
.36106
.50317
.60472
.68591
.75442
.81413
.86729
.91537
.95937
1
0
.15208
.30600
.43872
.54811
.64273
.72696
.80337
.87362
.93889
1
Table 3
Upper bounds on worst-case effectiveness with optimally chosen reserve
price in i.i.d. environments
α
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1
r(B),ciid
E4
(α) 0
.31415
.41496
.50583
.59351
.67969
.75400
.82260
.88612
.94509
1
r(B),ciid
E12
(α) 0
.45226
.56441
.65517
.73015
.79224
.84005
.88439
.92563
.96407
1
r(B),ciid
E25
(α) 0
.5883
.65178
.70938
.76190
.80998
.85416
.89489
.93255
.96750
1
Unfortunately, we cannot analytically identify the distributions that attain worst case
effectiveness for c.i.i.d. environments where the seller sets the reserve price optimally
ciid that
given his beliefs. The fact that, for example, in Table 2, on the environment in Bn,α
attains worst case effectiveness with no reserve price for n = 12 and α = 10%, choosing the
reserve price optimally increases effectiveness from .32411 to .89579, but for n = 12 and
α = 50%, choosing the reserve price optimally increases effectiveness only from .73769 to
.75282 suggests that setting the reserve price optimally may significantly improve worstcase effectiveness for environments with low α’s, but it may have only a negligible effect
on worst-case effectiveness for environments with larger α’s.
However, we are able to determine the worst-case effectiveness of the English auction
with an optimally chosen reserve price for i.i.d. environments that satisfy some regularity
condition as shown in Table 3 for values of n = 4, 12, and 25.15
15 The proof is quite involved and is not reproduced here. It can be obtained from the author upon request.
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
227
The fact that for α larger than 40%, the differences between the values depicted in
Tables 2 and 3 are below 10% demonstrates that, unless α is small, an optimally chosen
reserve price does not improve worst-case effectiveness by much on conditionally i.i.d.
environments.16
5. Comparison to the posted price mechanism
In this section we determine the worst-case effectiveness of the posted price mechanism.
As explained in the introduction, the motivation for this exercise is not to argue for the
superiority of the English auction over posted-prices, but rather to “calibrate” expectations
as to what constitutes “reasonable” worst-case performance.
As Milgrom (1989, p. 18) writes “Posted prices are commonly used for standardized,
inexpensive items sold in stores.” Several authors examined the relative performance of
posted-prices compared to auctions (see (Wang, 1993; Kultti, 1999) and the references
therein) and compared to bargaining (see (Wang, 1995) and the references therein) and
described conditions under which the posted-price mechanism may outperform, or at least
perform as well, as either auctions or bargaining.
As in the rest of the literature, we assume that posted prices are set optimally given the
seller’s beliefs about the distribution of the buyers’ valuations for the object. We have the
following result,
Theorem 5. For every n 2 and α ∈ (0, 1], the worst-case effectiveness of the posted-price
mechanism is given by
E PP (α) =
1
1 − log(ε)
(16)
where ε is the unique solution to ε(1 − log(ε)) = α in the interval (0, 1]. It is obtained
ciid ) where all the buyers have the same identical valuation that is
on the distribution (in Bn,α
distributed according to the truncated Pareto distribution Fε in (12).
Since English auctions with optimally chosen reserve prices obviously dominate
reserve prices alone which are equivalent to the posted-price mechanism, the more
interesting comparison is between the English auction with no reserve price and the postedprice mechanism. These two sale mechanisms perform better in very different types of
environments. The English auction with no reserve price performs relatively well when
bidders’ valuations are positively correlated, and relatively poorly when bidders’ valuations
are negatively correlated. In contrast, the posted-price mechanism performs relatively well
16 For larger α’s, the fact that the possibility of setting an optimal reserve price is not very valuable for the
seller is consistent with Bulow and Klemperer’s (1996) result that an English auction with no reserve price and
n + 1 bidders generates a higher expected revenue than an English auction with an optimally chosen reserve
price but n bidders. However, it should be emphasized that the two-step environments that attain worst case
effectiveness violate one of the conditions (specifically, downward-sloping MR) that is maintained throughout
Bulow and Klemperer’s analysis.
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Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
Table 4
Worst-case effectiveness of the posted-price mechanism
α
E PP (α)
0
0
.1
.20451
.2
.25036
.3
.29076
.4
.33088
.5
.37336
.6
.42080
.7
.47679
.8
.54813
.9
.65282
1
1
when buyers’ valuations are negatively correlated because by charging a high price, the
seller can extract more surplus from the buyer with the highest valuation, but it performs
relatively poorly in environments where the buyers’ valuations are positively correlated
because it cannot exploit the implied “competition” among the buyers.
We depict the value of E PP (α), which is independent of the number of buyers, n, in
Table 4.
In spite of the fact that, especially when we restrict our attention to c.i.i.d. environments,
the worst-case performance of the English auction is better than that of the posted-price
mechanism, the worst-case performance of the latter compares favorably with that of the
former for low n’s and α’s.
6. Discussion
The inspiration for this paper came from what has been called the “Wilson critique.”
Wilson emphasized that in contrast to optimal mechanisms that are tailored to specific
environments, the rules of real economic institutions “are not changed as the environment
changes; rather they persist as stable, viable institutions” (1987, p. 36). In Wilson (1985),
he argued that good economic institutions must not rely on features that are common
knowledge among the agents such as (in the context of auctions) the number of potential
bidders, the bidders’ and seller’s probability assessments (i.e., the prior), and the functional
form of the dependence of the bidders’ willingness to pay for the object on their types.
While asking that mechanisms be independent of whatever is commonly known among
the agents seems somewhat extreme, it is upheld by the fact that in “practical situations,”
little, if at all, is commonly known among the relevant agents. Wilson (1985) presented
the double-auction as a premier example of a simple institution, (obviously, the English
auction provides another such example), and demonstrated its Pareto incentive efficiency
when the number of buyers and sellers is large. We described some of the research that has
followed in the introduction.
Except for the literature on double-auctions mentioned above, the literature that is most
closely related to our work in its motivation is the one that identifies environments in which
simple mechanisms are optimal. The motivating idea is that if it can be demonstrated
that these environments are general enough, then the prevalence of simple mechanisms
is explained.17 In auction theory, the early work of Vickrey (1961) showed that the
most widely used auction forms such as the English, the Dutch, and the sealed-bid firstprice auctions are equivalent in terms of the expected revenues they generate for the
17 McAfee (1992, p. 284), for example, writes “Finding the restriction that leads to the optimality of simple
mechanisms . . . [is] the most important problem facing mechanism design.”
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
229
seller in private values environments in which the bidders’ valuations are independently
and identically distributed. Myerson (1981) then established the optimality of these
auction forms in these environments.18 More recently, Lopomo (1998) showed that an
“augmented” English auction (where the seller sets the reserve price optimally after all but
one of the bidders dropped out) maximizes the expected revenue for the seller among
all ex-post incentive compatible and ex-post individually rational auction mechanisms.
In contract theory, Holmstrom and Milgrom (1987), Laffont and Tirole (1987), and
McAfee and McMillan (1987) have established the optimality of linear incentive contracts
in specific classes of environments. In contrast to this literature that demonstrates the
optimality of simple mechanisms in special environments in order to explain their
prevalence in general environments, the approach taken here is to focus on one well-known
simple auction mechanisms—the English auction—and to show that while perhaps strictly
sub-optimal in most environments, it is nevertheless reasonably effective in a wide range
of plausible environments.
Finally, as for the merit of worst-case analysis: if there existed an agreed upon prior
over the set of all possible environments, a better indicator of the effectiveness of the
English auction could be given by its average (according to this prior) rather than its worstcase performance. The fact that such a prior does not exist indicates that the nature of
the uncertainty facing the seller is difficult to quantify in terms of (objective) risk. Under
such circumstances, worst-case analysis still allows us to form sensible judgements about
the quality of performance of the English auction in spite of the fact that discussion of
“average performance” is impossible.
Acknowledgments
I thank an anonymous referee, Itzhak Gilboa, Hsueh-Ling Huynh, Massimo Marinacci,
Gerhard Orosel, Steve Tadelis, and seminar audiences at Harvard/MIT, the Hebrew
University, Northwestern, Stanford, Technion, Yale, the Summer in Tel Aviv 1998, and
the Decentralization conference at NYU for their comments. Financial support from the
NSF under grant SBR-9806832 is gratefully acknowledged.
Appendix A. Proofs
r(1,α)
r(B )
We begin by identifying E1
(α) and E1 1 (α) (it is straightforward to see that E10 (α) = 0 for all
α ∈ (0, 1]). Auctions with only one bidder are not very interesting, but the results illustrate our method of proof
and will become useful later.
Lemma 1. For every α ∈ (0, 1],
r(1,α)
E1
(α) =
r(α − r)
(1 − r)α
where r = 1 −
√
1 − α.
18 However, Myerson (1981) followed by Crémer and McLean (1985, 1988) and McAfee and Reny (1992) also
showed that, even within the confines of the private values model, when the identical distribution and then the
independence assumptions are relaxed, the resulting optimal auctions are very different from any auction that is
used in practice.
230
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
Proof. Fix an α ∈ (0, 1] and a reserve price r ∈ [0, 1). We show that (1) the worst case effectiveness of the
English auction with reserve price r over the environments in B1,α is larger or equal to r(α − r)/((1 − r)α);
and (2) describe a family of distributions {Bεα }ε>0 ⊆ B1,α such that for every ε > 0, the effectiveness of
the English auction with reserve price r in the environment Bεα is equal to r(α − r + ε)/((1 − r + ε)α) r(α − r)/((1 − r)α), when ε 0. Finally, (3) we show that
√ the highest worse case effectiveness over the
environments in B1,α is obtained at the reserve price r = 1 − 1 − α.
(1) Consider any cumulative distribution function B that induces an expectation α. Since
1
α=
1 − B(x) dx =
0
r
1 − B(x) dx +
1
1 − B(x) dx r + (1 − r) 1 − B(r − ) ,
r
0
where B(r − ) = limr r B(r), it must be that B(r − ) (1 − α)/(1 − r). Therefore,
r(α − r)
1−α
=
RE (B, r) = r 1 − B(r − ) r 1 −
.
1−r
1−r
Now, since r(α − r)/((1 − r)α) is increasing in α, for every α ∈ (0, 1] the worst case effectiveness of the English
auction with reserve price r over the environments in B1,α is larger or equal to r(α − r)/((1 − r)α).
(2) For every α ∈ (0, 1], r ∈ [0, α) and ε > 0, define the cumulative distribution function Bεα,r ∈ B1,α as
follows: Bεα,r (x) = 0 for x ∈ [0, r − ε), Bεα,r (x) = (1 − α)/(1 − r + ε) for x ∈ [r − ε, 1) and Bεα,r (1) = 1
(suppose that ε < r so that Bεα,r is indeed a cumulative distribution function). It is straightforward to
verify that the effectiveness of the English auction with reserve price r in the environment Bεα,r is
r(α − r + ε)/((1 − r + ε)α) r(α − r)/((1 − r)α), when ε 0.
(3) Finally, for a given α ∈ (0, 1), we compute the reserve price r that generates the highest worse case effectiveness over B1,α . (When α = 1, r = 1 is the best reserve price.) That is, we compute arg maxr∈[0,1] {r(α − r)/
((1 − r)α)}. Since drd (r(α − r)/((1 − r)α)) = (r 2 − 2r + α)/((1 − r)2 α), the maximum is obtained at the
√
smaller root of r 2 − 2r + α, namely r = 1 − 1 − α. ✷
Lemma 2. For every α ∈ (0, 1],
E1r(B) (α) =
1
ε
=
1 − log(ε) α
where ε is the unique solution to ε(1 − log(ε)) = α in the interval (0, 1].
Proof. Fix an α ∈ (0, 1]. Consider any distribution function B ∈ B1,α that is not a truncated Pareto distribution
Fε as in (12) where ε ∈ [0, 1] is such that
1 ε(1 − log(ε)) = α. The fact that B is different from Fε and that both
1
belong to B1,α , i.e., 0 (1 − B(x)) dx 0 (1 − Fε (x)) dx = α implies that there must exist an x̂ ∈ (ε, 1] such that
B(x̂) < Fε (x̂). Therefore,
x̂(1 − B(x̂)) x̂(1 − Fε (x̂))
ε
maxr∈[0,1] {RE (B, r)} RE (B, x̂)
=
>
=
α
α
α
α
α
where the last equality follows from (13).
✷
The proofs of Theorems 1–3 proceed in two steps. First, in Lemma 3 below we show that among all
the distributions B ∈ Bn,α that induce the same distribution of the highest valuation G, the minimal ratios
En0 (α; x1 ∼ G), Enr(n,α) (α; x1 ∼ G), and Enr(B) (α; x1 ∼ G), are obtained at the distribution that induces the
distribution of the second highest valuation H ∗ that is described below. Then, we show that among all the
distributions B ∈ Bn,α that induce a distribution of the second highest valuation H ∗ , the minimal ratios
En0 (α; x2 ∼ H ∗ ), Enr(n,α) (α; x2 ∼ H ∗ ), and Enr(B) (α; x2 ∼ H ∗ ) are obtained on certain distributions of the highest
valuation that are described below.
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
231
Lemma 3. Among all the distributions B ∈ Bn,α that induce a distribution of the highest valuation G, the minimal
r(B)
ratios En0 (α; x1 ∼ G), and En (α; x1 ∼ G), are obtained on those distributions that induce a distribution of the
second highest valuation that is given by
G(x), for 0 x < x̂,
H ∗ (x) =
(17)
1,
for x̂ x 1,
1
1
where x̂ is such that 0 (1 − H ∗ (x)) dx = (nα − β)/(n − 1) and β = 0 (1 − G(x)) dx.
Proof. The proof formalizes the intuition that for any given reserve price r and distribution of the highest
valuation G, worst-case effectiveness is obtained on those distributions where the second highest valuation is
as small as possible (or, where H is “pushed” to the left as much as possible). Fix a reserve
1 price r ∈ [0, 1]
and a distribution B ∈ Bn,α that induces a distribution of the highest valuation G such that 0 (1 − G(x)) dx =
RFB (B) = β. Denote
1 second highest valuation by H .
1the distribution of the
We show that r (1 − H (x)) dx r (1 − H ∗ (x)) dx. This follows immediately for r x̂. We assume
valuations from among v1 , . . . , vn ,
therefore that r < x̂. Recall that x1 , . . . , xn denote the largest
n to smallest n
E[x
]
=
where the
latter
are
distributed
according
to
B.
Because
i
i=1
i=1 E[vi ] nα and E[x1 ] = β,
E[x2 ] + ni=3 E[xi ] nα − β, and since E[x2 ] E[xi ] for all i ∈ {3, . . . , n},
E[x2 ] nα − β
.
n−1
(18)
1
Since RE (Bn , 0)/RFB (Bn ) = E[x2 ]/β, the proof for En0 where r = 0 ends here. Suppose that r (1 − H (x)) dx <
1
∗
∗
r (1 − H (x)) dx where r < x̂. Because H is the distribution of x2 , H (x) G(x) = H (x) for all x ∈ [0, r].
Therefore,
r
E[x2 ] =
1 − H (x) dx +
=
0
1 − H (x) dx <
r
0
1
1
r
0
1 − G(x) dx +
1
1 − H ∗ (x) dx
r
nα − β
1 − H ∗ (x) dx =
.
n−1
A contradiction to (18). Therefore, for every r ∈ [0, 1] and distribution B ∈ Bn,α that induces a distribution of
1
highest valuation G such that 0 (1 − G(x)) dx = β,
1
1
r(1 − G(r − )) + r (1 − H ∗ (x)) dx
RE (B, r) r(1 − G(r − )) + r (1 − H (x)) dx
=
.
RFB (B)
β
β
The last inequality holds for every r ∈ [0, 1], and in particular, for the r that maximizes the last expression.
Therefore, among all the distributions B ∈ Bn,α that induce the distribution of the highest valuation G, the
r(n,α)
r(B)
minimal ratios En
(α; x1 ∼ G) and En (α; x1 ∼ G) are obtained at the distribution that induces a distribution
of the second highest valuation that is given by H ∗ . ✷
Proof of Theorem 1. By Lemma 3, we may restrict our attention to those distributions B ∈ Bn,α that induce a
distribution of the second highest valuation that is given by H ∗ . We show that among all such distributions, the
minimal ratio En0 (α; x2 ∼ H ∗ ) = max{(nα − 1)/(n − 1), 0} is obtained on a distribution where one (randomly
chosen) bidder has valuation 1 with probability min{nα, 1} and valuation 0 otherwise, and all other bidders have
valuations max{(nα − 1)/(n − 1), 0}.
Fix a distribution B ∈ Bn,α that induces a distribution of the highest valuation G and a distribution of the
second highest valuation H ∗ . Recall that E[x1 ] = β. By definition of H ∗ ,
E[x2 ] =
nα − β
.
n−1
Therefore, minimal effectiveness is equal to,
232
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
min
β∈[0,1]
nα − 1
E[x2 ]
nα − β
= max min
, 0 = max
,0
β∈[0,1] (n − 1)β
E[x1 ]
n−1
since (nα − β)/((n − 1)β) is decreasing in β.
✷
Proof of Theorem 2. Fix an n 2 and α ∈ (0, 1). By Lemma 3, we may restrict our attention to those
distributions B ∈ Bn,α that induce a distribution of the second highest valuation that is given by H ∗ . We show
that worst-case effectiveness among such distributions is given by

r(n, α)n(α − r(n, α))


, α 2 n(1 − α) 1,

(1 − r(n, α))(nα − (n − 1)r(n, α))
r(n,α)
∗
(α; x2 ∼ H ) =
(19)
En

nα − 1


,
α 2 n(1 − α) > 1,
n−1
√
where r(n, α) = α(n − n − nα)/(n − 1 + α) when α 2 n(1 − α) 1. When α 2 n(1 − α) > 1, every reserve price
in the interval [0, (nα − 1)/(n − 1]) is optimal. When α 2 n(1 − α) 1, worst-case effectiveness is obtained on
the limit of a sequence of distributions where one randomly chosen bidder which the seller cannot identify has
valuation 1 with probability (n(α − r(n, α)))/(1 − r(n, α)) and valuation just below r(n, α) otherwise, and all
other bidders have valuations just below r(n, α). When α 2 n(1 − α) > 1, worst-case effectiveness is attained on
the same environments on which En0 (α) is attained.
Fix some r ∈ [0, α].19 By (2) and Lemma 3, the worst-case effectiveness of any distribution B ∈ Bn,α that
1
induces a distribution of the highest valuation G with RFB (B) = 0 (1 − G(x)) dx = β is bounded from below by

nα − β

 r 1 − G(r − ) , r >
,
RE (B, r) 
n−1

nα − β
nα − β
β


,
r
.
n−1
n−1
By Lemma 1, there exists a sequence of distributions on the limit of which r(1 − G(r − )) attains its lower bound
of r(β − r)/((1 − r)β). It therefore follows that,

r(β − r)
nα − β


, r>
,
RE (B, r)  (1 − r)β
n−1
(20)

β
nα − β
nα − β


,
r
.
n−1
n−1
We are interested in the value of β ∈ [α, 1] for which the low bound (20) is the lowest. The fact that
r(β − r)/((1 − r)β) is increasing in β and that r(β − r)/((1 − r)β) r (nα − β)/(n − 1) implies that the
lowest value of (20) is obtained at the lowest value of β which satisfies r > (nα − β)/(n − 1), or at the limit
where β = min{nα − nr + r, 1} > α. It follows that for every r ∈ [0, α], worst-case effectiveness is bounded from
below by

nα − 1
rn(α − r)


, r
,

(1 − r)(nα − nr + r)
n−1
(21)

nα − 1
nα − 1


,
r
.
n−1
n−1
depending on whether the lowest possible β is obtained on nα − nr + r or 1.20 We now solve for the reserve
price r that maximizes (21). The fact that
n((n + α − 1)r 2 − 2αnr + nα 2 )
rn(α − r)
d
=
dr (1 − r)(nα − nr + r)
(1 − r)2 (nα − nr + r)2
19
20
It is staightforward to verify that for r > α worst-case effectiveness is equal to zero.
Note that nα − nr + r 1 if and only if (nα − 1)/(n − 1) r.
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
233
implies that as a √function of the reserve price r, rn(α − r)/((1 − r)(nα − nr√+ r)) is increasing on the
21
interval [0, α(n − n − nα)/(n − 1 + α)), and decreasing on the
√interval (α(n − n − nα)/(n − 1 + α), α).
Worst-case effectiveness is therefore obtained on r =√α(n − (n − nα))/(n − 1 + α) where it is equal to
rn(α − r)/((1 − r)(nα − nr + r)) provided that α(n − (n − nα))/(n − 1 + α) (nα − 1)/(n − 1), or on any
r ∈ [0, (nα − 1)/(n − 1)] where it equals (nα − 1)/(n − 1), otherwise. Finally, inspection of the inequality
√
α(n − n − nα ) nα − 1
n−1+α
n−1
reveals that for n 1 and α ∈ [0, 1], it is satisfied if and only if
√
1 + 4n − 3
0α
.
✷
2n
Proof of Theorem 3. By Lemma 3, we may restrict our attention to those distributions B ∈ Bn,α that induce a
distribution of the second highest valuation that is given by H ∗ . We show that among all such distributions, the
r(B)
minimal ratio En (α; x2 ∼ H ∗ ) = (nα − β)/((n − 1)β) is obtained on a distribution that induces a distribution
of the highest valuation that is a truncated Pareto distribution Fε as in (12) where ε = (nα − β)/(n − 1) and
β ∈ [0, 1] satisfies (11), and all other valuations are equal to ε.
1
By Lemma 2, for every distribution G with 0 (1 − G(x)) dx = β,
E1r(B) (β) =
maxr∈[0,1] {r(1 − G(r − ))} maxr∈[0,1] {r(1 − Fε (r))}
ε
=
β
β
β
where ε is such that ε(1 − log(ε)) = β. Therefore, for every distribution B ∈ Bn,α that induces a distribution of
1
the highest valuation G with 0 (1 − G(x)) dx = β,
1
maxr∈[0,1] {RE (B, r)}
maxr∈[0,1] {r(1 − G(r − ))} 0 (1 − H ∗ (x)) dx
max
,
β
β
β
ε nα − β
max
,
,
β (n − 1)β
where ε ∈ [0, 1] is such that
ε 1 − log(ε) = β.
(22)
(23)
The first term in the right-hand side of (22) corresponds to the revenue obtained from the bidder with the highest
valuation when the reserve price is r and the second term corresponds to the revenue obtained from the bidder
with the second highest valuation (i.e., when r = 0). Because, for ε ∈ (0, 1), ε(1 − log(ε)) is increasing in ε, ε and
therefore also ε/β = 1/(1 − log(ε)) is increasing in β. On the other hand, (nα −β)/((n − 1)β) is decreasing in β.
Therefore, minβ∈[α,1] {max{ε/β, (nα − β)/((n − 1)β)}} is obtained at a β that satisfies22
ε=
nα − β
n−1
(24)
and by plugging (17) back into (16) it follows that the minimum ratio is obtained at a β ∈ [0, 1] that satisfies
nα − β
nα − β
1 − log
= β.
✷
n−1
n−1
ciid
Proof of Theorem
n4. Fix an n 2 and an α ∈ (0, 1). Considern any belief B ∈ Bn,α . By assumption,
B(v1 , . . . , vn ) =
i=1 F (vi , z) dZ(z) for every (v1 , . . . , vn ) ∈ [0, 1] where for every z, F (· , z) ∈ B1,α(z) , and
√
√
The numbers α(n − n − nα)/(n − 1 + α) and α(n + n − nα)/(n − 1 + α) are
√ the two roots of the
2
2
equation (n + α − 1)r − 2αnr + nα = 0. It is staightforward to verify that α < α(n + n − nα)/(n − 1 + α).
Note also that rn(α − r)/((1 − r)(nα − nr + r)) = (nα − 1)/(n − 1) when r = (nα − 1)/(n − 1).
22 β α follows from the fact that when β = α, (nα − α)/((n − 1)α) = 1 > ε/α because ε/α =
1/(1 − log(ε)) < 1 since ε ∈ (0, 1].
21
234
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
α(z) dZ(z) α. We show that unless every F (· , z) is a “two-step” function as in (15), the minimal ratios
En0,ciid (α) and Enr(n,α),ciid (α) cannot be obtained on it. The idea of the proof is to show that unless every F (· , z)
ciid such that
∈ Bn,α
is a two-step function, the distribution B can be changed into another distribution B
r)
RE (B, r) RE (B,
>
RFB (B)
RFB (B)
(25)
ciid can be arbitrarily closely approximated by
for every r ∈ [0, 1). Because every distribution function B ∈ Bn,α
ciid
a distribution function in Bn,α that describes the distribution of n random variables that are i.i.d. conditional
on a random variable that obtains only finitely many values, it is sufficient to prove the theorem only for
ciid that may be
every such function. That is, we may restrict our attention to distribution functions B ∈ Bn,α
n
n where F ∈ B
c
F
(v
)
for
every
(v
,
.
.
.
,
v
)
∈
[0,
1]
written as B(v1 , . . . , vn ) = m
j
j
i
1
n
j
1,αj for every
j=1
i=1
m
j ∈ {1, . . . , m}, m
j=1 cj αj α, and (c1 , . . . , cm ) is a vector of positive weights such that
j=1 cj = 1. For every
such distribution function, the distribution of the highest and second highest bidders’ valuations are given by
G(v) =
m
n
cj Fj (v)
and
H (v) =
j=1
m
n−1
n cj n Fj (v)
− (n − 1) Fj (v) ,
j=1
for every v ∈ [0, 1], respectively.
j = Fj + δφ ∈ B1,αj for some αj ∈ (0, 1].
Let φ : [0, 1] → [−1, 1] be a smooth bounded function such that F
Application of (1) and (2), respectively, yields,
n j
=
RFB F
1
n 1 − Fj (x) + δφ(x) dx =
0
i=0
0
1
=
1 n
i n−i
n Fj (x) φ(x)
δ n−i
1−
dx
i
n 1 − Fj (x) dx − δn
0
1
Fj (x)
n−1
φ(x) dx ± O δ 2
0
= RFB (Fj )
n
1
− δn
Fj (x)
n−1
φ(x) dx ± O δ 2
(26)
0
where O(δ 2 ) denotes order of magnitude δ 2 ,23 and
n j , r = r 1 − Fj (r − ) + δφ(r − ) n
RE F
1
n−1
n 1 − n Fj (x) + δφ(x)
+ (n − 1) Fj (x) + δφ(x) dx
+
r
n n−1
= r 1 − Fj (r − ) − δnφ(r − ) Fj (r − )
1
n−1
n +
+ (n − 1) Fj (x) dx
1 − n Fj (x)
r
1
− δn(n − 1)
n−2 φ(x) Fj (x)
1 − Fj (x) dx ± O δ 2
r
n−1
= RE (Fj )n , r − δnφ(r − ) Fj (r − )
1
n−2 1 − Fj (x) dx ± O δ 2 .
− δn(n − 1) φ(x) Fj (x)
r
23
A function h(δ) is of an order of magnitude δ k , denoted O(δ k ), if limδ0 h(δ)/δ k is finite.
(27)
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
235
Lemma 4. For every constant C ∈ [0, 1], and every cumulative distribution function F ∈ B1,α , if F is not a twostep function as in (15), then there exist two numbers 0 a < b < 1 such that F (a) < F (b), and
(n − 1)((F (b))n−2 (1 − F (b)) − (F (a))n−2 (1 − F (a)))
= C.
(F (b))n−1 − (F (a))n−1
If r > 0, then numbers 0 a < r < b < 1 can also be chosen such that F (a) < F (b), and
(n − 1)(F (b))n−2 (1 − F (b))
= C.
(F (b))n−1 − (F (a))n−1
(28)
(29)
Proof. By definition, if F is not a two-step function then it assumes at least three different values on the interval
[0, 1). It follows that there exist three numbers 0 x1 < x2 < x3 < 1 such that 0 F (x1 ) < F (x2 ) < F (x3 ) 1.
For n = 2, the left-hand side of (28) is equal to −1 for any F (a) < F (b).24 Since for n 3 the left-hand side
of (28) is strictly decreasing in F (b) on the interval [0, 1], it must have at least two different valuations, and the
conclusion follows.
Suppose now that 0 < r < 1. If F is not a two-step function then there exist three numbers 0 x1 < x2 <
x3 < 1 where x1 < r, x2 = r, and x3 > r such that 0 F (x1 ) < F (x2 ) < F (x3 ) 1. Since the left-hand side of
(29) is strictly decreasing in F (b) on the interval [0, 1], we may choose 0 a < r < b < 1 to satisfy (29). ✷
Note that for every distribution function Fj ∈ B1,αj , and for every two numbers 0 a < b < 1, such that
Fj (a) < Fj (b), there exist two non-overlapping intervals of length l > 0, Ia and Ib , respectively, such that the
average value of Fj on Ia is Fj (a), and the average value of Fj on Ib is Fj (b).
Suppose now that r = 0 and that one of the Fj ’s is not a two-step function. Then, as the previous
lemma shows, two numbers a and b can be chosen to satisfy (28) for every constant C, and in particular for
C = RE (B, r = 0)/RFB (B). Distinguish between the following two possibilities:
(1) there exist two numbers 0 a < b < 1, such that
(n − 1)((Fj (b))n−2 (1 − Fj (b)) − (Fj (a))n−2 (1 − Fj (a))) RE (B, r = 0)
>
;
(Fj (b))n−1 − (Fj (a))n−1
RFB (B)
(30)
(2) there exist two numbers 0 a < b < 1, such that
(n − 1)((Fj (b))n−2 (1 − Fj (b)) − (Fj (a))n−2 (1 − Fj (a))) RE (B, r = 0)
<
.
(Fj (b))n−1 − (Fj (a))n−1
RFB (B)
(31)
In case (1), if Fj (a) > 0, then let φ : [0, 1] → [−1, 1] be a smooth function that approximates a function that is
equal to −1 on Ia , 1 on Ib , and zero otherwise; and if Fj (a) = 0, then let φ : [0, 1] → [−1, 1] be a smooth function
that approximates a function that is equal to 1 on Ib , and zero otherwise. In case (2), let φ : [0, 1] → [−1, 1] be a
smooth function that approximates a function that is equal to 1 on Ia , −1 on Ib , and zero otherwise. Note that in
j = Fj + δφ ∈ B1,αj for every small enough δ.
every case above, φ can be chosen so that F
It can be readily verified that for every four real numbers A, B > 0, and x, y 0, such that x < A and y < B,
A
A−x
<
B −y
B
⇔
x
A
> ,
y
B
(32)
and
A+x
A
x
A
<
⇔
< .
B +y
B
y
B
j ∈ B1,αj . Consider case (1). Note that,
j = Fj + δφ with δ > 0 small enough so that F
Define F
n
n
r = 0)
RE (B,
k=j ck RE ((Fk ) , r = 0) + cj RE ((Fj ) , r = 0)
=
n
n
RFB (B)
k=j ck RFB ((Fk ) ) + cj RFB ((Fj ) )
(33)
24 As will become clear at the completion of the proof, this implies that for n = 2, E 0,ciid (α) is obtained on
2
mixtures of i.i.d distributions with support on 0 and 1.
236
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
which by (26) and (27), is equal to,
1
m
n
n−2 (1 − F (x)) dx ± O(δ 2 )
j
j=1 cj RE ((Fj ) , r = 0) − δn(n − 1) 0 φ(x)(Fj (x))
.
1
m
n ) − δn
n−1 φ(x) dx ± O(δ 2 )
c
R
((F
)
(F
(x))
j
j
j=1 j FB
0
r = 0)/RFB (B)
is approximately equal to,
Therefore, by definition of φ, RE (B,
m
n
n−2 (1 − F (b)) − (F (a))n−2 (1 − F (a))) ± O(δ 2 )
j
j
j
j=1 cj RE ((Fj ) , r = 0) − δn(n − 1)((Fj (b))
m
.
n ) − δn((F (b))n−1 − (F (a))n−1 ) ± O(δ 2 )
c
R
((F
)
j
j
j
j=1 j FB
Finally, the fact that for small δ, terms of order of magnitude δ 2 may be ignored, together with (30) and (32),
imply (25). Consider now case (2). As before, by (26) and (27),
1
m
n
n−2 (1 − F (x)) dx ± O(δ 2 )
r = 0)
j
RE (B,
j=1 cj RE ((Fj ) , r = 0) − δn(n − 1) 0 φ(x)(Fj (x))
.
=
1
m
n
n−1
2)
RFB (B)
c
R
((F
)
)
−
δn
(F
(x))
φ(x)
dx
±
O(δ
j
FB
j
j
j=1
0
r = 0)/RFB (B)
is approximately equal to,
Therefore, by definition of φ, RE (B,
m
n
n−2 (1 − F (b)) − (F (a))n−2 (1 − F (a))) ± O(δ 2 )
j
j
j
j=1 cj RE ((Fj ) , r = 0) + δn(n − 1)((Fj (b))
m
.
n ) + δn((F (b))n−1 − (F (a))n−1 ) ± O(δ 2 )
c
R
((F
)
j
FB
j
j
j
j=1
As before, for small δ, terms of order of magnitude δ 2 may be ignored, together with (31) and (33), this implies
(25).
The proof for the case where r > 0 employs a similar idea. Suppose that r > 0 and that one of the Fj ’s is not
a two-step function. By Lemma 4 there exist two numbers a < r < b that satisfy (29) for every constant C, and
in particular for C = RE (B, r)/RFB (B). Distinguish between the following two possibilities:
(1) the two numbers 0 a < r < b < 1 are such that
(n − 1)(Fj (b))n−2 (1 − Fj (b)) RE (B, r)
>
;
(Fj (b))n−1 − (Fj (a))n−1
RFB (B)
(34)
(2) there exist two numbers 0 a < r < b < 1, such that
(n − 1)(Fj (b))n−2 (1 − Fj (b)) RE (B, r = 0)
<
.
(Fj (b))n−1 − (Fj (a))n−1
RFB (B)
(35)
In case (1), if Fj (a) > 0, then let φ : [0, 1] → [−1, 1] be a smooth function that approximates a function that is
equal to −1 on Ia , 1 on Ib , and zero otherwise; and if Fj (a) = 0, then let φ : [0, 1] → [−1, 1] be a smooth function
that approximates a function that is equal to 1 on Ib , and zero otherwise. In case (2), let φ : [0, 1] → [−1, 1] be a
smooth function that approximates a function that is equal to 1 on Ia , −1 on Ib , and zero otherwise. Note that in
j = Fj + δφ ∈ B1,αj for every small enough δ.
every case above, φ can be chosen so that F
j ∈ B1,αj . Consider case (1). Note that by definition
Define Fj = Fj + δφ with δ > 0 small enough so that F
r)/RFB (B)
is approximately equal to,
of φ, RE (B,
m
n
n−2 [1 − F (b)] ± O(δ 2 )
r)
j
RE (B,
j=1 cj RE ((Fj ) , r) − δn(n − 1)(Fj (b))
= m
.
n
n−1
c
R
((F
)
)
−
δn((F
(b))
−
(Fj (a))n−1 ) ± O(δ 2 )
RFB (B)
j
j
j=1 j FB
As before, the fact that for small δ, terms of order of magnitude δ 2 may be ignored, together with (34) and (32),
r)/RFB (B)
is approximately
imply (25). Consider now case (2). As before, the definition of φ implies that RE (B,
equal to,
m
n
n−2 (1 − F (b)) ± O(δ 2 )
r = 0)
j
RE (B,
j=1 cj RE ((Fj ) , r = 0) + δn(n − 1)(Fj (b))
m
=
n
n−1
− (Fj (a))n−1 ) ± O(δ 2 )
RFB (B)
j=1 cj RFB ((Fj ) ) + δn((Fj (b))
which, for small δ, together with (35) and (33), implies (25). Finally, we write that worst-case effectiveness is
obtained on a limit of a sequence of mixtures of i.i.d. two-step distributions because of reasons similar to those in
Z. Neeman / Games and Economic Behavior 43 (2003) 214–238
237
the proof of Lemma 1, namely, for similar considerations, the “jump” in F may occur “just before” the reserve
price r(n, α). ✷
Proof of Theorem 5. Fix an n 2 and an α ∈ (0, 1]. Consider any environment B ∈ Bn,α . Let G denote the
corresponding distribution of max{v1 , . . . , vn }. The expected revenue under the posted-price mechanism is given
by
max r 1 − G(r − )
r∈[0,1]
where G(r − ) = limxr G(x). The effectiveness of the posted-price mechanism in the environment B is therefore
given by
r(1 − G(r − ))
.
(36)
max 1
r∈[0,1]
0 (1 − G(x)) dx
1
Suppose that 0 (1 − G(x)) dx = β α. By Lemma 2, for every such β,
r(1 − G(r − ))
1
max 1
r∈[0,1]
1
−
log(ε)
0 (1 − G(x)) dx
where ε ∈ [0, 1] is the unique solution to ε(1 − log(ε)) = β. Because, for ε ∈ (0, 1), ε(1 − log(ε)) is increasing in
ε, ε and therefore also 1/(1 − log(ε)) is increasing in β. Therefore, the minimum of 1/(1 − log(ε)) is obtained at
β = α. In fact, the inequality in (36) is binding at the environment where all the buyers’ valuations are identical
and distributed according to the truncated Pareto distribution Fε where ε(1 − log(ε)) = α. It therefore follows
that E PP (α) = 1/(1 − log(ε)) where ε ∈ [0, 1] is the unique solution to ε(1 − log(ε)) = α. ✷
Calculating En0,ciid (α) and Enr(n,α),ciid (α). Fix some n 3 and α ∈ (0, 1]. By Theorem 4, we know that
En0,ciid (α) and Enr(n,α),ciid (α) are obtained on distributions that are mixtures of i.i.d. two-step distribution functions.
For every r ∈ [0, 1), we solve numerically for the particular two-step functions that attain worst-case effectiveness
on i.i.d. environments with n bidders and expected valuations αj , j ∈ J. Denote these functions by {Fαj }j∈J . As
noted in the text, all these two-step functions have a first step that is equal to zero. We then numerically solve for,
n
j∈J λj RE ((Fαj ) , r)
max min
n
r∈[0,1] {λj }j ∈J
j∈J λj RFB ([Fαj ] )
subject to
λj αj α,
j∈J
λj = 1,
j∈J
and
λj 0
for every j ∈ J.
As noted in the text, remarkably, the minimum is obtained on degenerate mixtures, namely, for every αj ∈ (0, 1],
the minimum is obtained on the vector {λk }k∈J where λj = 1 and λk = 0 for every k = j .
We also conducted robustness checks to verify that replacing the two-step distribution functions that attain
worst-case effectiveness for i.i.d. environments by other two-step distribution functions and minimizing subject
to the constraints above only increases the value of the objective function.
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